I'm solving some QFT problems and one of them deals with the Yukawa coupling. It asks to consider three processes, namely, $\psi\psi\to\psi\psi$, $\psi\bar{\psi}\to\psi\bar{\psi}$ and $\phi\phi\to\phi\phi$, and then find the Feynman diagrams to second order and derive the matrix elements.

The issue is that if I understood the coupling, there is one single vertex type and it requires a fermion particle, a fermion anti-particle and a scalar particle to meet at the vertex.

For the first and last processes I simply can't imagine a diagram with the given external legs, given number of internal points (namely one and two) and with the internal pointa subject to the constraint that a $\psi$ line a $\bar{\psi}$ line and a $\phi$ line must meet there. Not even the simple tree level diagrams for the $s,t,u$ channels seems to work.

How can I find the Feynman diagrams for the given processes in Yukawa theory? What is the correct reasoning and the correct diagrams?


I'm assuming you're using the lagrangian


or something similar.

The diagrams don't require you to have a particle-antiparticle-scalar triple to make a vertex. It just requires that all interaction vertices must have two $\psi$ propagators and one $\phi$ propagator. So long as the charge flow into each vertex is valid, you are fine. For instance, think of the first process,


as a $t$ channel process. It's just the


diagram, but rotated on its side (that is, related by a crossing symmetry).

The $\phi\phi\to\phi\phi$ process is actually a bit more complicated, and there isn't a diagram to second order that contributes to it. The lowest order diagram would be a box diagram (see below, with curvy lines representing $\phi$ propagators and straight lines representig $\psi$ propagators).

box diagram

I hope this helped!

  • $\begingroup$ Thanks, with this I've found for particle-particle scatering the $t,u$ diagrams with a relative sign between them. The $s$ diagram isn't there by charge conservation if I'm right. I didn't find any diagram with loops though, just tree level ones. There should be a diagram with a loop? Or the other diagram with loop will be the $\phi\phi\to\phi\phi$ process as you've outlined? $\endgroup$
    – Gold
    Jun 18 '17 at 21:29

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